# Jacobian conjecture

Let $F\colon\mathbb{C}^{n}\to\mathbb{C}^{n}$ be a polynomial map, i.e.,

 $F(x_{1},\dots,x_{n})=(f_{1}(x_{1},\dots,x_{n}),\dots,f_{n}(x_{1},\dots,x_{n}))$

for certain polynomials $f_{i}\in\mathbb{C}[X_{1},\dots,X_{n}]$.

If $F$ is invertible, then its Jacobi determinant $\det(\partial f_{i}/\partial x_{j})$, which is a polynomial over $\mathbb{C}$, vanishes nowhere and hence must be a non-zero constant.

The Jacobian conjecture asserts the converse: every polynomial map $\mathbb{C}^{n}\to\mathbb{C}^{n}$ whose Jacobi determinant is a non-zero constant is invertible.

Title Jacobian conjecture JacobianConjecture 2013-03-22 13:23:46 2013-03-22 13:23:46 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Conjecture msc 14R15 Keller’s problem